13,727 research outputs found
General limit value in Dynamic Programming
We consider a dynamic programming problem with arbitrary state space and
bounded rewards. Is it possible to define in an unique way a limit value for
the problem, where the "patience" of the decision-maker tends to infinity ? We
consider, for each evaluation (a probability distribution over
positive integers) the value function of the problem where the
weight of any stage is given by , and we investigate the uniform
convergence of a sequence when the "impatience" of the
evaluations vanishes, in the sense that . We prove that this uniform convergence happens
if and only if the metric space is totally bounded.
Moreover there exists a particular function , independent of the
particular chosen sequence , such that any limit point of such
sequence of value functions is precisely . Consequently, while speaking of
uniform convergence of the value functions, may be considered as the
unique possible limit when the patience of the decision-maker tends to
infinity. The result applies in particular to discounted payoffs when the
discount factor vanishes, as well as to average payoffs where the number of
stages goes to infinity, and also to models with stochastic transitions. We
present tractable corollaries, and we discuss counterexamples and a conjecture
The value of Repeated Games with an informed controller
We consider the general model of zero-sum repeated games (or stochastic games
with signals), and assume that one of the players is fully informed and
controls the transitions of the state variable. We prove the existence of the
uniform value, generalizing several results of the literature. A preliminary
existence result is obtained for a certain class of stochastic games played
with pure strategies
Continuous bounded cocycles
Let be a minimal locally compact groupoid with compact metrizable unit
space and let be a continuous -Hilbert bundle. We show that a bounded
continuous cocycle c: G\ra r^*E is necessarily a continuous coboundary. This
is a groupoid version of a classical theorem of Gottschalk and Hedlund.Comment: 14 pages, presented at the EU-NCG4 Conference, Bucharest 201
Random walks on Bratteli diagrams
In the eighties, A. Connes and E. J. Woods made a connection between
hyperfinite von Neumann algebras and Poisson boundaries of time dependent
random walks. The present paper explains this connection and gives a detailed
proof of two theorems quoted there: the construction of a large class of states
on a hyperfinite von Neumann algebra (due to A. Connes) and the ergodic
decomposition of a Markov measure via harmonic functions (a classical result in
probability theory). The crux of the first theorem is a model for conditional
expectations on finite dimensional C*-algebras. The proof of the second theorem
hinges on the notion of cotransition probability.Comment: 18 pages, written version of a talk given at the Operator Theory 26th
Conference, Timisoara 201
AF-equivalence relations and their cocycles
After a review of some of the main results about hyperfinite equivalence
relations and their cocycles in the measured setting, we give a definition of a
topological AF-equivalence relation. We show that every cocycle is cohomologous
to a quasi-product cocycle. We then study the problem of determining the
quasi-invariant probability measures admitting a given cocycle as their
Radon-Nikodym derivative.Comment: 15 pages, talk at 4th International Conference on Operator Algebras,
July 2-7 2001, Constanza, Romani
- …